Diagnosing extratropical synoptic-scale stratosphere-troposphere exchange: A case study

Q. J. R. Meteorol. SOC. (1999), 125, pp. 635-655

Diagnosing extratropical synoptic-scale stratosphere-troposphere exchange: A case study

By V. WIRTH* and J. EGGER University of Munich, Germany

(Received 25 February 1998; revised 2 September 1998)

SUMMARY Five methods of diagnosing the synoptic-scale mass exchange across the tropopause are investigated and

compared with each other. Three of them are derived from Wei’s general formula using pressure, potential temperature, and potential vorticity as vertical coordinates. The fourth method involves the computation of a large number of trajectories. The fifth method evaluates the flux directly as the difference between the motion of the air and the motion of the tropopause. The methods are tested in the context of a case study. An episode with strong convective heating in a decaying cut-off cyclone was selected. A consistent data set was obtained for this event through a special run of the European Centre for Medium-Range Weather Forecasts model with enhanced output storage.

There was a large cross-tropopause mass exchange during the episode. It is found that the different diagnostic methods yield quite different results. Those versions of Wei’s formula which predominantly draw on standard information like wind, temperature and pressure turn out to be unreliable as they suffer from strong cancellation of large terms. Tests with the direct technique reveal that the problem lies in the lack of consistency between the model data and the diagnosed location of the tropopause. The amount of cancellation is substantially reduced when full use is made of model information about the non-conservative fields in Wei’s general formula with potential vorticity as vertical coordinate. This formulation also allows a straightforward interpretation of the net flux in terms of individual physical processes. The trajectory method, which does not require information about the non-conservative fields, also yields reasonable estimates for the cross-tropopause mass flux. The problems with the different methods and their differences show up especially clearly in a second model run in which all non-conservative processes are switched off.

KEYWORDS: Cut-off cyclone Mass flux Stratosphere-troposphere exchange Tropopause

1. INTRODUCTION

The issue of stratosphere-troposphere exchange has traditionally been viewed as a problem of synoptic- and smaller-scale dynamics, with attention focused on the phe- nomenology of strong mixing events such as mid-latitude tropopause folds and deep tropi- cal convection (WMO 1986). More recently the emphasis has shifted away from individual events at the tropopause towards the global picture in which the stratospheric zonal mean circulation plays a major role (Holton et al. 1995). Although this ‘change in paradigm’ leads to a conceptual simplification in a number of ways, there is observational evidence suggesting that the lowest part of the extratropical stratospheric circulation cannot simply be described in terms of a zonal-mean circulation (Holton et al. 1995; Tuck et al. 1997). In particular, the distribution of trace gases, such as water vapour and ozone, cannot be explained without accounting for zonally asymmetric synoptic-scale events. The latter, which influence the lowermost stratosphere from below and sideways, appear to be important for a complete understanding of the chemistry of this part of the atmosphere (Bregman et al. 1997; Lelieveld et al. 1997).

The synoptic-scale extratropical stratosphere-troposphere exchange remains, therefore, a relevant issue, and it is important to have a detailed and quantitative knowledge concerning the spatial and temporal distribution of the mass exchange across the tropopause. In contrast to the zonal-mean exchange across the 100 hPa surface which can be diagnosed using the ideas of ‘downward control’ (Holton 1990; Haynes et al. 1991; Rosenlof and Holton 1993), the synoptic-scale exchange across the tropopause cannot be obtained within this framework. For this purpose, various versions of a diagnostic formula derived by Wei (1987) have been used in the past. In particular, Wei’s formula has been applied

* Corresponding author: Meteorologisches Institut, Theresienstrde 37,80333 Munchen, Germany.

635

636 V. WIRTH and J. EGGER

to model data from global numerical models in order to learn how the hemispheric mass exchange is distributed in space and time (Hoerling et al. 1993; Grewe and Dameris 1996).

In this paper we investigate and compare various different methods that allow the diagnosis of the extratropical synoptic-scale stratosphere-troposphere exchange. It is shown that certain versions of Wei’s formula, which have been used extensively in the past, may yield rather unreliable estimates for the cross-tropopause mass flux even when optimistic assumptions are made concerning data availability and resolution. The reasons for the apparent shortcomings are discussed in some detail.

To address the issue outlined above, we have studied a particular case of a cut-off cyclone, which occurred in June 1994 over the Mediterranean and was accompanied by strong deep convection. Wirth (1995a), hereafter W95, has suggested that the diabatic heating in such a situation may not only lead to the gradual decay of the upper-level circulation, but may also be responsible for a substantial mass exchange from the stratosphere to the troposphere. While the results of W95 are certainly suggestive, they were only based on an axisymmetic mechanistic model. The present case was selected because its synoptic-scale evolution corresponds well with the idealized scenario in W95. As a first step it is shown that, indeed, the diabatic mechanism from W95 is responsible for a substantial cross-tropopause mass exchange in the current example. Therefore, the case is considered to be well suited for the investigation of different methods of diagnosing this mass exchange (the main topic of this study).

The paper is organized as follows. First we define the tropopause and introduce various formulae for diagnosing the cross-tropopause mass flux in section 2. Section 3 gives information about the model data used in this study, and section 4 presents the synoptic evolution of the case. In section 5 it is shown that, indeed, the selected case corresponds well to the idealized scenario of W95. The cross-tropopause mass flux during the episode considered is diagnosed in section 6 and various different methods are applied and critically compared with each other. A second data set, which stems from a purely conservative model run, is analysed in section 7 to highlight the differences between the various methods. Finally, section 8 provides a summary and conclusions.

2. TROPOPAUSE DEFINITION AND FORMULAE FOR DIAGNOSING THE CROSS-TROPOPAUSE

Previous research has shown that a threshold on potential vorticity (PV) yields a suitable definition of the extratropical tropopause in most circumstances (Hoskins et al. 1985; Hoerling et al. 1991). In the context of the primitive equations, PV is given by

MASS EXCHANGE

1 P

P = - 6, * ve

where p is the density, 6 is the potential temperature, 6, = f k + V x v is the absolute vorticity, f is the Coriolis parameter, k is the unit vector in the vertical, and v is the horizontal wind (Hoskins et al. 1985). The theshold values found in the literature lie typically between 1.6 and 3.5 PVU* (e.g. WMO 1986; Hoeriing et al. 1991; Grewe and Dameris 1996; Cox et al. 1997). Throughout this paper we use this so-called ‘dynamical tropopause’, i.e. we refer to the PV in order to define the tropopause.

* Potential vorticity units: 1 PVU = 10-6m2 s-’K kg-’

DIAGNOSING STRATOSPHERE-TROPOSPHERE EXCHANGE 637

z I

tr

1,,.......,

Figure 1. Schematic drawing depicting various quantities used in connection with the definition of the cross- tropopause mass flux. The solid line labelled z'p(x, y , t ) represents the tropopause at time 1, and the dashed line

labelled z'P(x, y , t - Dt) is the tropopause a short time earlier.

For conservative (i.e. adiabatic and frictionless) conditions the dynamical tropopause is simply advected by the flow. This immediately follows from

D P 1 1 ~ = - h * V Q + - V x X-VO Dt P P

(e.g. Hoskins et al. 1985), which expresses the material rate of change of PV in terms of the diabatic heating Q = (DO/Dt),,, and the non-conservative mechanical forcing (like friction) X = ( D V / D ~ ) ~ ~ ~ ~ . Here, D/Dt = a / & + u - V is the material derivative following the three-dimensional flow u.

The tropopause is regarded as a two-dimensional manifold in three-dimensional space, which changes its location with time. We assume that the geometric height of the tropopause z t P ( x , y , t ) can be expressed as a unique function of horizontal position (x, y ) and time t . Although this excludes multiple tropopauses and, hence, tropopause folds, it does not impose any restriction in the current context. From fundamental kinematical principles the flux of mass across the tropopause from the troposphere to the stratosphere is given by

(3) Here, n is the (upward) normal vector to the tropopause, and ctp represents the motion of the tropopause (see Fig. 1). The vector clp is defined to be parallel to n with its modulus given by lctpl = cos a l a z ' P / a t I , where a is the angle between the normal vector n and the vertical (a is assumed to be smaller than n/2) . The quantity ?(p) measures the amount of mass crossing the tropopause per unit time and unit tropopause area. More commonly, the cross-tropopause mass flux is expressed as a flux per unit horizontal area. The latter is denoted by F ( p ) and is obtained by accounting for the local slope of the tropopause,

A

F(P) = P (u - Ctp) ' n .


yielding

The flux F ( x ) of a tracer with mixing ratio x is obtained from (4) through simply replacing

Although formula (4) is very basic and its implementation seems straightforward (see P by XP.

section 6(b)), it is rarely used in practical applications. Instead, the formula

is quite popular. Here, q is a generalized vertical coordinate with J,, = az/aq, and q'p(x, y , t) denotes the location of the tropopause in terms of this coordinate. All terms on the right-hand side have to be evaluated at the local tropopause. Formula (5) was first derived by Wei (1987) by considering an atmospheric column divided into two sub-volumes by the tropopause and extending from the earth's surface to the top of the atmosphere. There is an alternative derivation of (3, which simply establishes the equivalence between (4) and (5) . It essentially involves the splitting of the the dot product in (4) into a vertical and a horizontal component and the introduction of a generalized vertical coordinate. The new derivation, which is given in the appendix, reveals that the terms in ( 5 ) are generally not directly related to the terms in (4). The last feature contributes to the difficulty of interpreting Wei's general formula in terms of physical processes (see below).

Depending on the choice of the vertical coordinate, one obtains different versions of Wei's formula. Choosing geometric height z as vertical coordinate, one obtains

Here, w = Dz/Dt denotes the vertical component of the wind. For the special case v = 0 it is easily verified that (6) is equivalent to (4). In a hydrostatic atmosphere with pressure p as vertical coordinate one obtains

were g is the gravity acceleration and w = Dp/Dt is the vertical wind in pressure coordinates. This version was used, for instance, by Grewe and Dameris (1996) and by Siegmund et al. (1996). Another version, which is frequently used (see, for example, Wei (1987), Hoerling et al. (1993), Ebel et al. (1996), and in a modified application also Spaete et al. (1994)), is obtained with potential temperature 8 as vertical coordinate:

Finally, formula (5 ) becomes particularly simple if one defines the tropopause via PV and, at the same time, uses PV as vertical coordinate (Hoerling et al. 1993):

az DP ap Dt

F(p ) = p - __ . (9)

In this case, the second and third terms on the right-hand side of (5 ) are zero, since the partial derivatives of PV following the PV-definedtropopause vanish trivially. It was argued


in Wirth (1995b) that (9) is particularly suitable for interpretation as it avoids the notorious cancellations occurring in most of the other versions of Wei's formula and, in combination with (2), it allows a clear attribution of the mass flux to the various (non-conservative) physical processes.

We introduce the obvious notation Wei-z, Wei-p, Wei-8 and Wei-PV for the different versions (6) to (9) of Wei's general formula (5). Note that each version requires different information as input. For the application of formula (6), for instance, one needs to know the three-dimensional wind and the location of the tropopause (and its temporal behaviour). Formula (8) only requires the horizontal wind instead of the three-dimensional wind, but in return one needs information about the diabatic heating Q = DB/Dt at the tropopause. Formula (9) requires knowledge about the non-conservative mechanical forcing X in addition to the diabatic heating Q, if one wants to use it in connection with (2). The different versions are also quite distinct in terms of interpretation. It was argued by Wirth (1995b) (see also Wei (1995)) that Wei-PV (9) in combination with (2) yields a physically mean- ingful interpretation of the mass flux in terms of physical processes, while this is generally not the case for other versions of (5).

3. MODELDATA

In order to produce a consistent data set, we have made use of the global numerical weather prediction model from the ECMWF* (Simmons et al. 1989). The version used was operational in early 1995. A special 3-day model run was carried out. It was initialized at 12 GMT 27 June 1994 and covered the whole period of interest for our case study. The model run was performed with a spatial resolution of T106 with 31 model levels and full model physics. The agreement between the model forecast and the verifying 500 hPa analyses is very satisfying (Fig. 2). In the following we assume that the model gives a realistic representation of the synoptic-scale evolution during this particular episode.

The model data were stored every three hours. For analysis they were interpolated onto a regular 1" x 1" grid in the horizontal and 34 pressure levels in the vertical, resulting

* European Centre for Medium-Range Weather Forecasts

Figure 2. Comparison of (a) the 48-hour model forecast and (b) the verifying analysis for 12 GMT 29 June 1994. Both plots show 500 hPa geopotential height (full lines, contours every 4 dam) and temperature (dashed lines,

contours every 2 K).


in a vertical resolution of A p = 25 hPa in the tropopause region. The data set thus produced is used as the primary source of information for the case study.

There are seven non-conservative physical processes that are treated by the ECMWF model in a parametrized manner: diabatic heating due to radiation (el); diabatic heating due to vertical diffusion (Qz); diabatic heating due to cumulus convection ( Q3); diabatic heating due to large-scale condensation (&); vertical diffusion of the horizontal wind (X,); gravity-wave drag (X,); and the convective tendency of the horizontal wind (X,). In the special model run, these fields were stored every three hours. They were, therefore, directly available for subsequent off-line analysis. This is in contrast to many practical applications where one often has to resort to rather sophisticated methods in order to obtain essential information indirectly from the data (e.g. Hoerling et al. 1993; Schaack et al. 1990).

4. SYNOPTIC DESCRIPTION OF THE CASE

The episode selected for the case study took place in June 1994 over the Mediterranean (Fig. 3). On 25 June 1994 there was a pronounced trough over western Europe on the 500 hPa geopotential height map, extending across the Bay of Biscay to Spain. This trough evolved into a cut-off cyclone over southern France on 26 June, which subsequently moved

Figure 3 . European Centre for Medium-Range Weather Forecasts analyses on four consecutive days in 1994: (a) 12 GMT 25 June, (b) 12 GMT 26 June, (c) 12 GMT 27 June, and (d) 12 GMT 28 June. The plots show the 500 hPa

geopotential height (full lines, contours every 4 dam) and temperature (dotted lines, contours every 2 K).

DIAGNOSING STRATOSPHERE-TROPOSPHERE EXCHANGE 64 1

30. 70. 100. 150. 200. 250. 300. 400. 500. 600. 700. 775.

875. 925. \

\

\

101 0. 5 7.5 10 12.5 15 17.5 20

Longitude

Figure 4. Vertical section along 42.5"N at 12 GMT 28 June 1994. The dashed lines represent potential temperature 0 (contours every 5 K), and the full lines represent potential vorticity P (contours at 0.5, 1, 1 .5 ,2 ,4 ,6 ,8 , and 10

P W ) . The area with P > 2 PVU is marked by stippling whose intensity increases with potential vorticity.

eastward over the Gulf of Genoa and reached Italy on 28 June. Later, the cut-off weakened and moved off in a south-easterly direction.

Figure 4 shows a vertical section through the cyclone on 28 June. In this plot, the centre of the vortex is characterized by a lowered tropopause, with a cold anomaly in the troposphere and a warm anomaly in the lower stratosphere. The cyclonic wind (not shown) reached a maximum of about 30 m s-lclose to the tropopause. Maps of the lower- tropospheric temperature (not shown) indicated that the vortex had an almost equivalent- barotropic structure during the whole episode, as there was very little phase tilt with height.

In the region of the cut-off cyclone, the short-term ECMWF forecast produced strong precipitation (Fig. 5) , the magnitude of which was approximately verified by individual measurements (such as, for example, the Berliner Wetterkarte). During 26 June and 27 June most of the precipitation fell on the eastern and northern flank of the cyclone (compare Fig. 5(a) with Fig. 3(c)). In particular, there was strong precipitation over land (south- eastern France and northern Italy). Only when the cyclone was located right over Italy on 28 June, was there a region of strong precipitation close to its centre. Since the latter is a distinctive feature of the mechanism studied in W95, we will focus on 28 June in the following, and especially on the time interval between 09 and 15 GMT, since on 28 June the bulk of the precipitation over Italy took place during this 6-hour period.

A separate analysis of large-scale and convective precipitation in the model data reveals that most of the precipitation by far was of convective nature. This is consistent with the form and intensity of the clouds on the satellite images, which suggests that strong deep convection occurred, especially over land (Fig. 6).

The tropopause is defined through P = 4 PVU. At first sight, this PV value appears somewhat high, but a lower value ( P = 2 or 3 PVU) turns out to be unsuitable in the present


Figure 5. Total precipitation (in mm) accumulated over 24 hours: (a) 00 to 24 GMT 27 June 1994 and (b) 00 to 24 GMT 28 June. The total precipitation is the sum of large-scale precipitation and convective precipitation. The contours are at 1,5, 10,20 and 50 mm and stippling indicates areas with more than 1 mm total accumulated

precipitation.

Figure 6. METEOSAT infrared satellite image at 1155 GMT 28 June 1994. Regions with deep convection appear as white areas.


case as it would result in an indistinct tropopause with arather smooth vertical PV gradient. This is probably a result of the 'convective erosion' during the preceding evolution. In the centre of the cut-off cyclone, the tropopause pressure decreased by about 30 hPa between 09 and 15 GMT on 28 June 1994. Using a fully Lagrangian trajectory method (to be described in more detail in subsection 6(c)) we found that the air, too, moved to smaller pressures in the centre of the cyclone, but that the pressure drop of individual parcels was only about 10 hPa. This is qualitatively consistent with the behaviour dicussed in W95; there was upward vertical air motion in the vortex centre, but the tropopause moved upwards even faster, such that the motion of the air with respect to the tropopause was downward.

In summary, the cut-off cyclone considered here is relatively close to the idealized vortex from W95 concerning its basic synoptic-scale features. The current case appears, therefore, suitable for the investigation of the stratosphere-troposphere exchange which, according to W95, accompanies the diabatic decay of such a cut-off cyclone.

5. STRATOSPHERE-TROPOSPHERE EXCHANGE THROUGH CONVECTIVE HEATING

Figure 7 shows a sequence of PV maps on an upper-tropospheric isentrope. Clearly, the cut-off cyclone appears as an isolated area of air with high (stratospheric) PV, surrounded by air with low (tropospheric) PV. The area enclosed by stratospheric PV decreases with

b I

Figure 7. Maps of potential vorticity P (full lines) on the 0 = 325 K isentrope every 12 hours: (a) 12 GMT 27 June 1994, (b) 00 GMT 28 June 1994, (c) 12 GMT 28 June 1994, (d) 00 GMT 29 June 1994. The contour interval is 1 PVU and the smallest value contoured is 1 PVU. Regions with P > 2 PVU are marked by stippling whose

intensity increases with potential vorticity.


- 70. - 100. - 150.

200. 250. 300. 400. 8 500. 8 600. 2 700. 9

825. - 875. 925. 975. 101 0.

775. 9

7.5 10 12.5 15 17.5 20 Longitude

Figure 8. Time-mean material rate of change of potential vorticity, D P / D t , between 06 and 18 GMT 28 June 1994 in a section through the cut-off cyclone along 45"N, i.e. somewhat to the north of its centre. The contours are drawn every 0.5 PVU (12 h)-', and the zero contour is suppressed; stippling indicates positive values over 0.5

PVU (12 h)-'. The thick solid line represents the tropopause at 12 GMT, defined by P = 4 PVU.

time, as does the maximum value of the PV. Both suggest the presence of non-conservative processes (see section 2)-possibly including the effect of numerical diffusion.

With the help of (2) , the mean material rate of change DP/Dt during a specific time interval was calculated, accounting for all seven non-conservative processes which are represented in the ECMWF model ( Qi , i = 1, 2, 3,4, and Xi, i = 1, 2, 3, see section 3). The model directly provides time-mean values of these fields over the time interval between time steps; the other quantities in (2) were linearly interpolated to the mid point of the time interval. The result for a 12-hour interval is displayed in Fig. 8 as a vertical section. There is a shallow volume of strong material PV decrease in the neighbourhood of the tropopause. The latter is depicted as a thick solid line. By considering each of the non-conservative processes individually it was verified that the heating due to cumulus convection yields by far the largest contribution to the pronounced extremum in Fig. 8; the contributions from the other processes are much smaller by comparison. It follows that the strong material decay of PV in the tropopause region is mainly a result of the synoptic-scale convective heating and its particular spatial structure.

The cross-tropopause mass flux F(p) was obtained from DP/Dt according to (9) by multiplying by p az/a P and interpolating to the tropopause. The temporal resolution used was 3 hours. An algebraic mean between the two end points of the 3-hour time interval is taken, both for the factor p a z / a P and for the location of the tropopause. The result for a 6-hour interval, i.e. the sum of two 3-hour intervals, is shown in Fig. 9(a). There was a strong downward flux (i.e. negative values of F ( p ) ) over northern Italy close to the centre of the cut-off cyclone, and much smaller values of F ( p ) elsewhere. Thanks to the use of Wei-PV (9) and the above analysis of DP/Dt, it can be concluded that the majority of this cross-tropopause mass flux was related to the convective heating in the deep convection,


0" 1 O T 20"E 1O"E 20"E

Figure 9. Time-mean mass flux across the tropopause between 09 and 15 GMT 28 June 1994 calculated using Wei-PV (9) (see text). In (a) all non-conservative processes are included in the computation, while in (b) only the diabatic heating due to cumulus convection is accounted for. The tropopause is defined through P = 4 PVU (the contour interval of the mass-flux isolines is 20 hPa d-', with the zero contour suppressed and negative values

below -20 hPa d-] marked by stippling, indicating an exchange from the stratosphere to the troposphere).

TABLE 1. ANALYSIS OF THE CROSS-TROFQPAUSE TIME-MEAN AREA-AVERAGED MASS EXCHANGE (IN 10' kg S-') IN THE CONTEXT OF THE CASE STUDY USING DIFFERENT

DIAGNOSTIC METHODS (SEE TEXT)

Wei-PV Wei-p Wei-0 Direct Trajectory Wei-PV-CC

Netexchange -1.9 -1.1 -2.1 -0.5 -1.8 -1.7 Downward -2.5 -3.3 -4.4 -3.0 -2.6 -2.0 Upward 0.6 2.2 2.3 2.5 0.8 0.3

The area considered extends from 40" to 50" in latitude and from 5"E to 20"E in longitude and encompasses the whole cut-off cyclone; the time interval considered is 9 to 15 GMT 28 June 1994. The first row gives the net exchange, which results after horizontal averaging as a residuum of areas with downward mass flux (negative values, second row) and other areas with upward mass flux (positive values, third row).

very much as postulated in W95. Figure 9(b) explicitly illustrates the predominance of the diabatic mechanism in the present case; it shows the contribution to the mass flux due to convective heating only, which is obviously quite similar to the full mass flux in Fig. 9(a).

Integrating the mass flux displayed in Fig. 9(a) over the area between 40"N and 50"N and between 5"E and 2WE, one obtains an average mass exchange of -1.9 x lo9 kg s-' during the time interval between 9 and 15 GMT. This net exchange results from an integration over areas with a downward mass transfer (totalling -2.5 x lo9 kg s-') and other areas with a partly compensating upward transfer (totalling +0.6 x lo9 kg s-l). These numbers are given in the first column (called 'Wei-PV') of Table 1. The component of the mass exchange due to diabatic heating from cumulus convection alone is given in the last column ('Wei-PV-CC'). According to these numbers, some 2.0/2.5 = 80% of the downward mass transfer across the tropopause is accounted for by convective heating.

It is instructive to perform a quantitative comparison of our results with the mechanistic model study W95. In the latter, the mass transfer was purely downward and integration


over the entire idealized cut-off cyclone yielded a value of -2.8 x lo9 kg s-l. This is in fairly good agreement with the results from the present case study, both concerning the order of magnitude of the mass exchange and the fact that it is predominantly downward.

6. DIAGNOSING THE MASS EXCHANGE

A mass exchange of some lo9 kg s-l during a synoptic event can be considered as substantial, since this is close to the estimated magnitude of the annual-mean extratropical hemispheric exchange, which lies in the range 1.3 x lo9 to 12 x lo9 kg s-l (Follows 1992). In this sense, the episode between 9 and 15 GMT 28 June is regarded as a strong event. Different methods for diagnosing the mass flux should yield similar results. It turns out that this is not the case.

(a) Different versions of Wei ’s formula So far we have diagnosed the cross-tropopause mass flux with the help of Wei-PV

(9) in connection with (2). This has been relatively straightforward since, in our special model run, we stored the non-conservative fields in addition to the standard information like wind, geopotential height and temperature. In many practical applications, however, the non-conservative fields are not given as part of the data set, and other versions of Wei’s formula appear more favourable.

The result of the mass-flux calculation using Wei-p (7) and Wei-0 (8) is shown in Fig. 10(a) and (b), respectively, with the corresponding area-averaged values given in the second (‘Wei-p’) and third column (‘Wei-0’) of Table 1. As before, model data are used every three hours. More specifically, we took 3-hour averages for the non-conservative fields and for w, centred differences for the time derivative, and algebraic-mean values of the two end points of the time interval for the other quantities. Note that all terms required for the computation are explicitly taken from the model output. This especially concerns terms like w or D0/Dt, which in many practical applications are unknown and have to be obtained indirectly. Despite the availability of a complete and consistent data set, the fields displayed in Fig. 10(a) and (b) appear somewhat suspect. In both cases the maximum of the (downward) mass flux is located over the sea (to the west of Italy), rather than over land where most of the convection was observed (cf. Fig. 5(b)). In addition, both methods yield areas with substantial upward mass transfer, which was not the case with Wei-PV used in the previous section (Fig. 9(a)). Correspondingly, for Wei-p and Wei-0 the area integrated net transfer is composed of fairly large, and partly compensating, upward and downward contributions (see Table 1). The near agreement in the net exchange between Wei-8 and Wei-PV must be considered fortuitous, since the two methods differ significantly in their individual components (upward and downward, respectively).

A closer analysis of the results from Wei-p and Wei-0 reveals that the individual terms in the respective formulae are large and that there is a great deal of (locally varying) near compensation. This suggests that the fields shown in Fig. 10(a) and (b) are at least partly a numerical artefact rather than a real mass flux. As was shown by Siegmund et al. (1996) in the case of Wei-p, the problem of near compensation is particularly bad when there is horizontal advection of steep tropopause gradients. The problem is quite similar to that of Wei-0. Both formulae yield unreliable results when the temporal resolution is too coarse to match the horizontal resolution. According to Siegmund et al. (1996), the different character of the various terms in a finite-difference representation calls for a more sophisticated quasi-Lagrangian treatment.


0" 1 0 5 20"E 0" 1 0"E 20"E I f I

1 O"E

........ . .

Figure 10. As Fig. 9, except that the method used for the diagnosis is (a) Wei-p (Eq. (7)), (b) Wei-8 (Eq. (8)), (c) the direct method, and (d) the trajectory method (see text).

(b) Direct method Apparently, in our case the use of of Wei-p and Wei-8 is quite problematic. One may

ask whether the direct use of the basic formula (4), which was shown to be equivalent to Wei's formula, is any better.

The implementation of (4) is called the 'direct method'. It is done by approximating the tropopause through a numer of triangles (two triangles for each horizontal grid box). Consider three points defining one such triangle (Fig. 11). The normal vector n and the angle a! in (4) are represented by the respective quantity of the triangle, and the wind field u and the density p are approximated by their respective value at the centre of gravity s of the triangle (obtained through linear interpolation). The motion of the tropopause ctp is calculated from the difference between two points which lie both on the straight line along n through s and on the tropopause at the next available later and earlier times, respectively.

The mass flux thus obtained is shown in Fig. lO(c). In many ways, the result is similar to the result of Wei-p (Fig. 10(a)) and Wei-8 (Fig. lO(b)). Although the algorithm of computation of the direct method is completely different from the other two methods,


n \

I

\

\ \

I \ Figure 1 1 . Schematic diagram illustrating the implementation of the ‘direct method’. There are three triangles locally approximating the tropopause at time t - Dt, t and t + Dt, respectively. The dashed linen is perpendicular to the triangle z&), the point s (marked by *) is the centre of gravity of the triangle z w ( t ) , and the two crosses

(x) denote the point of intersection between n , zp(t - Dt) and zQ(t + Dt), respectively.

the patterns in Figs. 10(a), (b) and (c) resemble each other, in particular concerning their smaller-scale features. Also, in all three methods the net flux is composed of rather large and nearly compensating upward and downward contributions (Table 1 second, third, and fourth columns). This suggests that the problem of near compensation in all three methods goes back to a common origin. In the following, the problem is investigated in more detail using the direct method as this is the most suitable for physical interpretation.

Analysis of the individual terms in (4) shows that the smaller-scale features with large and opposing values of F ( p ) in close proximity to each other arise from the locally varying near compensation of terms in the evaluation of u . n. The computation of this dot product turns out to be problematic since the three-dimensional wind u is approximately parallel to the tropopause. It follows that the value, and even the sign, of the term u . n depends sensitively on the accuracy and consistency of the numerical representation of its factors. Since the tropopause (and, hence, the normal vector n) is not part of the consistent model data set but instead is obtained by an a posteriori interpolation between pressure levels, there is no guarantee of a high degree of consistency. Apparently the sensitivity is large enough to preclude a reasonable result with the given spatial and temporal resolution of the data. The computation of the term ctp, on the other hand, appears less problematic, although it may still suffer from the defect noted by Siegmund et al. (1996).

(c ) Trajectory method In contrast to Wei-p, Wei-8 or the direct method, Wei-PV does not suffer from the

problem of strong cancellation, at least as long as the non-conservative fields are known with reasonable accuracy. It is, therefore, not surprising that Wei-PV yields a result which looks more reasonable than those of Wei-p, Wei-8 and the direct method. Nevertheless, it would be comforting to verify the ‘good’ results derived from Wei-PV using yet another independent method. This is accomplished with the so-called ‘trajectory method’, which is


a Lagrangian method based on the fundamental relation (4). One follows the motion of the air in the neighbourhood of the tropopause and compares this motion with the motion of the tropopause itself. Lamarque and Hess (1994), for instance, used’such a method in their case study. In contrast, however, we have implemented this method in a fully Lagrangian manner. At the initial time 3600 points are homogeneously distributed on the tropopause over the region of interest. The points are advected by the three-dimensional wind with a time step of 10 minutes. The wind information is obtained through linear interpolation in time and space from the three-hourly 1 O x 1 model data. Every three hours a reinterpolation onto a regular horizontal grid is performed in order to avoid too strong an accumulation or dilution of points in certain areas. At the end of the time period considered (6 hours in our case) the points indicate the location of the tropopause in the hypothetical situation of purely conservative flow. The pressure difference between this hypothetical tropopause and the actual tropopause is a direct measure for the exchange due to non-conservative processes. Errors with this method arise from the temporal and spatial interpolation of the wind and from the three-hourly reinterpolation, which introduces some smoothing.

The result of our trajectory method is shown in Fig. 10(d) and in the fifth column of Table 1. There is fair agreement with the corresponding results from Wei-PV, in particular concerning the partitioning of the net exchange into an upward and a downward contribution. In any case, even if partly fortuitous (see below) the agreement between Wei-PV and the trajectory method is considerably better than the agreement between either of these two methods on the one hand, and Wei-p, Wei-8 or the direct method on the other hand. The distribution of the mass flux computed from Wei-PV and the trajectory method (Figs. 9 and 10(d)) approximately reflects the distribution of the model precipitation (Fig. 5(b)) and is, therefore, consistent with the expectation resulting from W95. All this can be seen as evidence that Wei-PV and the trajectory method are able to diagnose the cross-tropopause mass exchange for the event investigated in a way that is qualitatively realistic and quantitatively broadly correct.

The apparent success of the trajectory method appears comprehensible for the following reasons. The component of the three-dimensional wind across the tropopause, u . n, is always rather uncertain no matter which method is used. However, the trajectory method does not explicitly rely on the knowledge of the term u . n, but rather on the knowledge of u at the location of the advected points. The wind information accumulates over the 3-hour interval to result in the new location of the advected point, which is compared directly with the location of the tropopause at the final horizontal position of the trajectory. The distance a point travels away from the tropopause during a time interval is less sensitive to how well the tropopause is diagnosed than is the direction of the wind with respect to the tropopause at a particular instant. Thus, at the expense of computing a large number of trajectories, the trajectory method circumvents a major problem that the direct method and those versions of Wei’s formula based on p and 8 have difficulties with.

7. CONSERVATIVE RUN

To provide further insight into the strengths and weaknesses of the various methods, we have also applied them in akinematically similar, but physically very different, situation. More specifically, we have performed a model run identical to the run analysed so far (our ‘control’ run) in every respect, except that all parametrized non-conservative processes (see section 3) are switched off. As a consequence, the evolution of the cut-off cyclone and its strength after three days (on 30 June) deviated slightly from the corresponding evolution and strength in the control run, but on 28 June the overall flow was still quite similar in both runs. On the other hand, the lack of the parametrized non-conservative processes in


0" 1O"E 20"E 1WE 20"E

W 1 0% 20"E 0" 1O"E 20"E ,

......... +-' (d) ....................................... .................

Figure 12. As Fig. 9, but for the 'conservative run' (see text).

the conservative run had, of course, major consequences for the cross-tropopause mass flux (section 2).

It is important to note that even a so-called conservative run is not stricty conservative, since there is always explicit horizontal diffusion (necessary to make the numerics stable) and implicit numerical diffusion (from finite differencing) in the model. Nevertheless, the net mass exchange in the conservative run should be much smaller than in the control run since the convective heating, which was shown to yield the major contribution in the latter, was switched off in the former. We want to find out to what degree this is reflected in the various methods of the mass flux diagnosis.

Figure 12 and Table 2 give the results for the conservative run in the same format as before. The small scale features apparent in Wei-p, Wei-8 and the direct method (Figs. 12(a), (b), and (c), respectively) are almost as strong as the typical fluxes in the control run (see also the last row in Table 2). Again, there is a striking similarity of patterns between the direct method and Wei-p (and partly also Wei-8). There is less compensation and there are much fewer small-scale features with the trajectory method (last column in Table 2 and Fig. 12(d), respectively), and Wei-PV yields, trivially, F(p ) = 0 (first column in Table 2, but no figure shown). For the same reasons as in the previous section we interpret the small-

DIAGNOSING STRATOSPHERE-TROPOSPHERE EXCHANGE 65 1

TABLE 2. ANALYSIS OF THE CROSS-TROWPAUSE MASS EXCHANGE (lo9 kg s-I) FOR THE CONSERVATIVE RUN (OTHER DETAILS AS TABLE 1)

Wei-PV Wei-p Wei-0 Direct Trajectory ~~

Net exchange 0 1.4 -0.3 -0.5 -0.6 Downward 0 -3.0 -2.3 -2.4 -1.3 Upward 0 4.4 2.0 1.9 0.7

scale details in Figs. 12(a), (b), and (c) as artefacts arising from the near compensation of serveral large terms.

The difference between the results of Wei-PV and the trajectory method in the conservative run brings up an important issue. Apparently, these two methods yield significantly different results, although neither of them suffers from the problem of near cancellation and the appearance of small-scale features. The essential point is that they diagnose slightly different quantities and that, therefore, one should not even expect perfect agreement. Wei-PV diagnoses quite directly the exchange due to the parametrized non-conservative processes, i.e. the seven processes represented in the ECMWF model (see section 3). This may, in fact, be the quantity one is interested in; explicit and implicit model diffusion, which are primarily a numerical necessity but hardly represent physical processes, are ignored. On the other hand, the trajectory method accounts for all non-conservative processes operating on the resolved scales, including implicit model diffusion. In the model, PV is not materi- ally conserved, even in the conservative run, and air parcels cross the tropopause leading to a non-zero cross-tropopause mass flux. In this sense the trajectory method is likely to provide a more reliable estimate of the mass exchange that takes place in the model. Which method gives a more faithful representation of the ‘mass flux’ depends on one’s point of view. Furthermore, neither method can possibly account for features which are smaller in scale than those resolved by the model. Finally, the difference between Wei-PV and the trajectory method in our conservative run (Table 2) indicates that the almost perfect quantitative agreement in the non-conservative run (Table 1) is somewhat fortuitous. Con- sidering the fact that the two methods measure slightly different quantities, one should not expect perfect agreement.

8. SUMMARY AND CONCLUSIONS

This study has dealt with the extratropical synoptic-scale stratosphere-troposphere exchange in the framework of a case study. A consistent set of data was obtained through a special run of the ECMWF model with standard resolution (T106 with 31 model levels), but enhanced (3-hourly) output storage. Five different methods of diagnosing the cross- tropopause mass flux were implemented and compared critically with each other. The tropopause was defined through a specific value of potential vorticity (PV) throughout the study.

The episode under investigation took place in June 1994, when a cut-off cyclone moved eastward across the western Mediterranean. On 28 June, which was during the mature stage of the cut-off cyclone, strong deep convection was observed close to its centre. We focused on this particular day because the synoptic situation corresponded well in its basic features with the idealized scenario in W95.

Using a method called Wei-PV (which later turned out to be one of the more reliable ones) we found firstly that there was a substantial exchange of mass from the stratosphere to the troposphere within a few hours, and secondly that this mass exchange was mostly due


to the convective heating and its particular spatial distribution. Moreover, the computation of a large number of trajectories revealed that the air in the tropopause region was displaced upward; however, owing to the action of diabatic heating the tropopause was displaced upward even faster, resulting in an effective downward mass exchange. All the above features were essential ingredients of the mass exchange in the mechanistic study W95. The present case study, therefore, suggests that the diabatic mechanism investigated in W95 can indeed be a viable mechanism in the real atmosphere.

It is likely that the magnitude of the mass exchange diagnosed for the specific 6-hour period under investigation is larger than one should expect when the exchange is averaged over the whole decay phase of a cut-off cyclone. First, in a typical cut-off cyclone the bulk of the precipitation probably occurs more at the periphery (as on 27 June) than close to its centre. Furthermore, the strong mass flux diagnosed in the current case for 28 June only lasts for about a quarter of a day, while during the rest of the day it is much smaller. We refrain from speculating about the importance of the diabatic mechanism in a climatological sense.

The presence of a strong and physically well understood exchange process makes the selected case well suited for an investigation of different methods of diagnosing the cross-tropopause mass flux. It turns out that using Wei’s (1987) formula with pressure p or potential temperature 8 as the vertical coordinate (called Wei-p and Wei-8, respectively) yields horizontal distributions of the cross-tropopause mass flux characterized by strong small-scale features of opposite signs in close proximity to each other. Correspondingly the net flux integrated over the entire cut-off cyclone is represented as a small residuum of locally varying upward and downward fluxes. These small-scale features do not have any apparent connection with the non-conservative physical processes in the model. Moreover, they arise from the near cancellation of serveral large terms in the respective formulae. It is concluded that they are numerical artefacts.

Wei’s general formula (our Eq. (5)) was shown to be equivalent to the basic formula (4), which expresses the cross-tropopause mass flux as the difference between the motion of the air and the motion of the tropopause. We have suggested a novel diagnostic method that directly implements (4), with the tropopause being approximated by a number of triangles. This ‘direct method’ is computationally completely different from any of the other methods considered here. It turns out to be rather unsatisfactory. One major problem is the computation of the term u . n, i.e. the component of the wind across the tropopause. Because the wind blows approximately parallel to the tropopause, the magnitude, and even the sign, of this term depends sensitively on the accuracy and consistency of wind and tropopause data. The latter are apparently inadequate, given the resolution of the data used.

Despite this lack of success, the direct method has helped us gain a deeper understanding of the difficulties with some versions of Wei’s formula. The distribution of the mass flux calculated using the direct method shows patterns that are very similar to the case of Wei-p (and, to a slightly lesser degree, also to those obtained with Wei-0). All three methods suffer from near compensation of several large terms giving rise to spurious small-scale features. Analysis of our direct method, which is most amenable to physical interpretation, suggests that the common origin of these difficulties is insufficient accuracy and consistency of data, in particular concerning the orientation of the tropopause surface. This problem cannot be overcome by transformation to a generalized vertical coordinate.

Neither the direct method nor Wei-p require information about the non-conservative processes. Wei-PV on the other hand, which does not suffer from the defects of Wei-p or the direct method, makes full use of the non-conservative fields. This suggests that the im~eased use of information about diabatic processes leads to a better result concerning the


cross-tropopause mass flux. In accordance with this interpretation the mass flux obtained from Wei-8, which makes partial use of the non-conservative fields from the model output, is somewhat more realistic than Wei-p and the direct method, but still less realistic than Wei-PV.

Unlike our study, many practical applications do not have the non-conservative fields available as part of their data sets. In such cases the application of Wei-8 or Wei-PV requires these fields to be computed indirectly from the available data. It is certainly possible that the errors thus incurred might deteriorate the results obtained from Wei-PV to such extent that the relative advantage of Wei-PV over those methods which do not make explicit use of the diabatic fields vanishes.

We also implemented another independent method that was Lagrangian and involved the computation of a large number of trajectories. This so-called trajectory method does not require the knowledge of the non-conservative fields. It is computationally expensive, but it does not suffer from the problems of Wei-p, Wei-8 and the direct method. Consequently, we found good agreement between the trajectory method and Wei-PV, even though the high level of quantitative agreement was partly fortuitous. Concerning the present case, we believe that the results from either of these two methods give a fairly realistic estimate of the magnitude of the cross-tropopause mass flux and its spatial distribution.

One important difference between our two successful methods (Wei-PV and the trajectory method) was clearly revealed through a model run in which all parametrized non- conservative processes were switched off. While in such a situation Wei-PV yields a strictly zero cross-tropopause mass flux, this is not the case with the trajectory method since the latter implicitly accounts for the non-conservation of PV due to explicit horizontal and implicit numerical diffusion. The trajectory method probably gives a more reliable estimate of the mass flux that actually takes place in the model, even though part of this flux is due to model diffusion that does not necessarily represent a physical process present in the real atmosphere.

Overall we conclude that, firstly there are cases in which the diabatic heating in a cut-off cyclone leads to substantial mass exchange from the stratosphere to the troposphere. Secondly the diagnosis of the synoptic-scale cross-tropopause mass flux from global model data as they are currently available is far from trivial; even with fairly high temporal resolution (3-hourly data for a 1" x 1" horizontal resolution) those versions of Wei's general formula which make little use of information about the diabatic fields show strong numerical artefacts. It appears that, in such a situation, broadly realistic results can only be obtained when more information about the physical processes is available or when a more expensive Lagrangian method is used.

ACKNOWLEDGEMENTS

Part of this study was done in the context of a 'special project' at the E C M W and we gratefully acknowledge its support. One of us (VW) would like to express his gratitude towards E. Klinker, without whose help it would have been much more difficult to carry out this work. The constructive remarks of two anonymous reviewers helped us to improve the presentation of several items.

APPENDIX

A new derivation of Wei's formula We wish to prove that Wei's general formula (5) follows from the basic kinematic

relation (4). The starting point is Eq. (4). As before, it is assumed that the tropopause is


given through ztp(x, y, t). First, the normal vector n is split into a vertical and horizontal component,

n=n, +nh. (A.1) Here, n, = cos a k and nh = - cos a vhztP, where vh is the horizontal gradient operator. Inserting these relations into (4) yields

which is equivalent to Wei-z, i.e. version (6) of Wei’s general formula. The second step consists of introducing a generalized vertical coordinate q ,

z = Z ( X , y7 r , t> , (A.3)

with J,, = a Z / a r . The location of the tropopause in the new coordinate system is denoted by $‘; it is implicitly given through

(A.4) Z t P k y , t> = Z b , y , r’p(x, y , t) , t } . The partial derivatives of z‘p are transformed according to

vhztp = vhz Jq VhVtp , a z t p a p az a t a t a t __ = J,,- + - .

Inserting this, together with

into (A.2) yields

After some cancellations one obtains

which is equivalent to Wei’s formula, equation (5).

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Diagnosing extratropical synoptic-scale stratosphere-troposphere exchange: A case study